As Jamal observed in the other ticket, the options recording the details of the saved templates was wrong, the ID of the stored template was not of the template post type.
I don't know how a template might have been saved with the wrong post type, but I notice you have two plugins installed for switching post types, so I suspect it may have been accidentally updated.
In any case, I simply deleted the option, which fixes the errors. I then created a new template for Books which I was able to save without issue, so it looks like the problem is resolved.
I watched the video, and I see that the fix failed on your server.
And went back to your server and did some more testing.
On inspecting the database I saw that the problem actually occurs on your site because the templates, which are saved as posts, are missing a value for post_type (which should be 'view-template').
When I manually set the value, the template showed up in the list and worked as normal.
When I edited it and re-saved it, the post type was lost again.
Well, as I said before, you have two post type switcher plugins on your site, and "Custom Post Type Changer" was active.
So I disabled the plugin, reset the value for the content template post type in the database, and lo and behold, everything works as expected.
I also tested creating a template for books, no problem.
I couldn't see any settings pages for that plugin. I suggest you stop using it, and maybe use the other more mature plugin created by a WP core contributor instead.
